Talk:Dym equation

Alternative formulation?

The Dym equation

$u_{t}=u^{3}u_{xxx}$

appears to be written in many places as

$u_{t}=(u^{-1/2})_{xxx}$

and it is claimed for example in

• Olver, Peter J. (1993). Applications of Lie groups to differential equations, 2nd ed. New York, NY; London: Springer-Verlag. ISBN 0387940073.

that the two forms are equivalent. Can an expert confirm that? Oleg Alexandrov (talk) 04:10, 26 January 2007 (UTC)

Well, Peter Olver is an überexpert so his word is law, but for your edification: Substitute $u=v^{-1/2}$ in $u_{t}=v^{3}u_{xxx}$ to get $(v^{-1/2})_{t}=v^{-3/2}(v^{-1/2})_{xxx}$ and do the differentiation with respect to t to get $-(1/2)v_{t}=(v^{-1/2})_{xxx}$. Finally, do some rescaling to get rid of the factor $-1/2$; for instance, set $v(t,x)=w(-(1/2)t,x)$. Any mistakes in my computations are left as an exercise for the reader. -- Jitse Niesen (talk) 08:40, 26 January 2007 (UTC)

Thanks! Frankly I was not expecting an answer to the above, it was more of a note to myself. (but big brother Jitse is watching. :)

I added the alternative formulation in the article (that one, $u_{t}=(u^{-1/2})_{xxx}$ is actually used much more often from what I saw in my google books search). If you have more to add to the article, you are more than welcome to. Oleg Alexandrov (talk) 16:47, 26 January 2007 (UTC)

Harry-Dym equation

Is this the same thing as the Harry-Dym equation, or is that something else? 67.198.37.16 (talk) 22:48, 16 September 2020 (UTC)